Camera Parameters - PowerPoint PPT Presentation

Camera Parameters

CS485/685 Computer Vision

Prof. George Bebis

• The physical image planeis the CCD array of n x m rectangular grid of photo-sensors.
• The pixel image plane(frame buffer) is an array of N x M integer values (pixels).

• The position of the same point on the image plane will be different if measured in CCD elements (x, y) or image pixels (xim, yim).

(assuming that the origin in both cases is the upper-left corner)

(xim, yim) measured in pixels

(x, y) measured in millimeters.

• Five reference frames are needed in general for 3D scene analysis.
• Object
• World
• Camera
• Image
• Pixel

• 3D coordinate system: (xb, yb, zb)
• Useful for modeling objects (i.e., check if a particular hole is in proper position relative to other holes)
• Object coordinates do not change regardless how the object is placed in the scene.

Our notation: (Xo, Yo, Zo)T

• 3D coordinate system: (xw, yw, zw)
• Useful for interrelating objects in 3D

Our notation: (Xw, Yw, Zw)T

• 3D coordinate system: (xc, yc, zc)
• Useful for representing objects with respect to the location of the camera.

Our notation: (Xc, Yc, Zc)T

• 2D coordinate system: (x f , y f )
• Describes the coordinates of 3D points projected on the image plane.

Our notation:(x, y)T

• 2D coordinate system: (c, r)
• Each pixel in this frame has integer pixel coordinates.

y

Our notation:(xim, yim)T

x

• In general, the world and camera coordinate systems are not aligned.

center of projection

optical axis

Y

X

Z

x

y

• To simplify mathematics, let’s assume:

(1) The center of projection coincides with the origin of the world coordinate system.

(2) The optical axis is aligned with the world’s z-axis and

x,y are parallel with X, Y

center of

projection

(3) Avoid image inversion by assuming that the image plane is in front of the center of projection.

(4) The origin of the image plane is the principal point.

center of

projection

• The model consists of a plane (image plane) and a 3D point O (center of projection).
• The distance f between the image plane and the center of projection O is the focal length(e.g., the distance between the lens and the CCD array).

center of

projection

• The line through O and perpendicular to the image plane is the optical axis.
• The intersection of the optical axis with the image plane is called principal point.

Note: the principal point is not necessarily the image center.

Y

X

Z

1

1

1

1/f

• Using matrix notation:
• Verify the correctness of the above matrix
• homogenize using w = Z

or

• Many-to-one mapping
• The projection of a point is notunique
• Any point on the line OP has the same projection

• Scaling/Foreshortening
• Object’s image size is inversely proportional to the distance of the object from the camera.

• When a line (or surface) is parallel to the image plane, the effect of perspective projection is scaling.
• When an line (or surface) is not parallel to the image plane, the effect is foreshortening (i.e., perspective distortion).

• Effect of focal length
• As f gets smaller, more points project onto the image plane (wide-angle camera).
• As f gets larger, the field of view becomes smaller (more telescopic).

• What happens to lines, distances, angles and parallelism?
• Lines in 3D project to lines in 2D (with an exception …)
• Distances and angles are notpreserved.
• Parallel lines do notin general project to parallel lines due

to foreshortening (unless they are parallel to the image plane).

• Vanishing point:
• Parallel lines in space project perspectively onto lines that on extension intersect at a single point in the image plane called vanishing point(or point at infinity).
• The vanishing point of a line depends on the orientation of the line and not on the position of the line.

Note: vanishing

points might lie

outside of the

image plane!

• Alternative definition for vanishing point:
• The vanishing point of any given line in space is located at the point in the image where a parallel line through the center of projection intersects the image plane.

• Vanishing line:
• The vanishing points of all the lines that lie on the same plane form the vanishing line.
• Also defined by the intersection of a parallel plane through the center of projection with the image plane.

vanishing

line

• The projection of a 3D object onto a plane by a set of parallel rays orthogonal to the image plane.
• It is the limit of perspective projection as

• Using matrix notation:
• Verify the correctness of the above matrix (homogenize using w=1):

• Parallel lines project to parallel lines.
• Size does not change with distance from the camera.

• Approximate perspective projection by scaled orthographic projection (i.e., linear transformation).
• Good approximation if:

(1) the object lies close to the optical axis.

(2) the object’s dimensions are small compared to its average distance from the camera

• The term is a scale factor now (e.g., every point is scaled by the same factor).
• Using matrix notation:
• Verify - homogenize using

• Camera and world coordinate systems have been aligned (i.e., all distances are measured in the camera’s reference frame).
• The origin of the image plane is the principal point.

• In general, world and pixel coordinates are related by additional parameters such as:
• the position and orientation of the camera
• the focal length of the lens
• the position of the principal point
• the size of the pixels

• Extrinsic: the parameters that define the location and orientation of the camera reference frame with respect to a known world reference frame.
• Intrinsic: the parameters necessary to link the pixel coordinates of an image point with the corresponding coordinates in the camera reference frame.

• Describe the transformation between the unknown camera reference frame and the known world reference frame.
• Typically, determining these parameters means:

(1) find the translation

vector that maps the camera’s

origin to the world’s origin.

(2) find the rotation matrix

that aligns the camera’s axes

with the world’s axes.

RT, T

R, -T

• Using the extrinsic camera parameters, we can find the relation between the coordinates of a point P in world (Pw) and camera (Pc) coordinates:

or

where RiTcorresponds to the i-th row of the rotation matrix

• Characterize the geometric, digital, and optical characteristics of the camera:

(1) the perspective projection (focal length f ).

(2) the transformation between image plane coordinates and pixel coordinates.

(3) the geometric distortion introduced by the optics.

(1) From Camera Coordinates to Image Plane Coordinates:

perspective projection:

(2) From Image Plane Coordinates to Pixel coordinates

(ox, oy) are the coordinates of the principal point

e.g., ox= N/2, oy= M/2 if the principal point is

the center of the image

sx, sycorrespond to the effective size of the pixels in the horizontal and vertical directions (in millimeters)

• Using matrix notation:

f/sy

f/sx

(3) Relating pixel coordinates to world coordinates

or

Image distortions due to optics

(1) Radial distortion:

k1, k2, and k3 are intrinsic parameters

Image distortions due to optics

(2) Tangential distortion:

p1 and p2 are intrinsic parameters

• The matrix containing the intrinsic camera parameters

(not including distortion parameters for simplicity):

• The matrix containing the extrinsic camera parameters:

• Using homogeneous coordinates:
• M is called the projection matrix(i.e., 3 x 4 matrix).

• Warning: homogenization is required to obtain the pixel coordinates:

• Assuming ox= oy= 0 and sx= sy= 1
• Verify:

M  Mp

Homogenize:

√

M  Mwp

where is the centroid of the object

(i.e., average distance from the camera)

• Verify:

√

Homogenize: